3.86 \(\int \frac{\cot ^7(c+d x)}{(a+a \sin (c+d x))^4} \, dx\)

Optimal. Leaf size=135 \[ -\frac{\csc ^6(c+d x)}{6 a^4 d}+\frac{4 \csc ^5(c+d x)}{5 a^4 d}-\frac{7 \csc ^4(c+d x)}{4 a^4 d}+\frac{8 \csc ^3(c+d x)}{3 a^4 d}-\frac{4 \csc ^2(c+d x)}{a^4 d}+\frac{8 \csc (c+d x)}{a^4 d}+\frac{8 \log (\sin (c+d x))}{a^4 d}-\frac{8 \log (\sin (c+d x)+1)}{a^4 d} \]

[Out]

(8*Csc[c + d*x])/(a^4*d) - (4*Csc[c + d*x]^2)/(a^4*d) + (8*Csc[c + d*x]^3)/(3*a^4*d) - (7*Csc[c + d*x]^4)/(4*a
^4*d) + (4*Csc[c + d*x]^5)/(5*a^4*d) - Csc[c + d*x]^6/(6*a^4*d) + (8*Log[Sin[c + d*x]])/(a^4*d) - (8*Log[1 + S
in[c + d*x]])/(a^4*d)

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Rubi [A]  time = 0.0856235, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2707, 88} \[ -\frac{\csc ^6(c+d x)}{6 a^4 d}+\frac{4 \csc ^5(c+d x)}{5 a^4 d}-\frac{7 \csc ^4(c+d x)}{4 a^4 d}+\frac{8 \csc ^3(c+d x)}{3 a^4 d}-\frac{4 \csc ^2(c+d x)}{a^4 d}+\frac{8 \csc (c+d x)}{a^4 d}+\frac{8 \log (\sin (c+d x))}{a^4 d}-\frac{8 \log (\sin (c+d x)+1)}{a^4 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^7/(a + a*Sin[c + d*x])^4,x]

[Out]

(8*Csc[c + d*x])/(a^4*d) - (4*Csc[c + d*x]^2)/(a^4*d) + (8*Csc[c + d*x]^3)/(3*a^4*d) - (7*Csc[c + d*x]^4)/(4*a
^4*d) + (4*Csc[c + d*x]^5)/(5*a^4*d) - Csc[c + d*x]^6/(6*a^4*d) + (8*Log[Sin[c + d*x]])/(a^4*d) - (8*Log[1 + S
in[c + d*x]])/(a^4*d)

Rule 2707

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[(x^p*(a + x)^(m - (p + 1)/2))/(a - x)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& EqQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{\cot ^7(c+d x)}{(a+a \sin (c+d x))^4} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^3}{x^7 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^2}{x^7}-\frac{4 a}{x^6}+\frac{7}{x^5}-\frac{8}{a x^4}+\frac{8}{a^2 x^3}-\frac{8}{a^3 x^2}+\frac{8}{a^4 x}-\frac{8}{a^4 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{8 \csc (c+d x)}{a^4 d}-\frac{4 \csc ^2(c+d x)}{a^4 d}+\frac{8 \csc ^3(c+d x)}{3 a^4 d}-\frac{7 \csc ^4(c+d x)}{4 a^4 d}+\frac{4 \csc ^5(c+d x)}{5 a^4 d}-\frac{\csc ^6(c+d x)}{6 a^4 d}+\frac{8 \log (\sin (c+d x))}{a^4 d}-\frac{8 \log (1+\sin (c+d x))}{a^4 d}\\ \end{align*}

Mathematica [A]  time = 0.170585, size = 89, normalized size = 0.66 \[ \frac{-10 \csc ^6(c+d x)+48 \csc ^5(c+d x)-105 \csc ^4(c+d x)+160 \csc ^3(c+d x)-240 \csc ^2(c+d x)+480 \csc (c+d x)+480 \log (\sin (c+d x))-480 \log (\sin (c+d x)+1)}{60 a^4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^7/(a + a*Sin[c + d*x])^4,x]

[Out]

(480*Csc[c + d*x] - 240*Csc[c + d*x]^2 + 160*Csc[c + d*x]^3 - 105*Csc[c + d*x]^4 + 48*Csc[c + d*x]^5 - 10*Csc[
c + d*x]^6 + 480*Log[Sin[c + d*x]] - 480*Log[1 + Sin[c + d*x]])/(60*a^4*d)

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Maple [A]  time = 0.148, size = 130, normalized size = 1. \begin{align*} -8\,{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{{a}^{4}d}}-{\frac{1}{6\,{a}^{4}d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{4}{5\,{a}^{4}d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{7}{4\,{a}^{4}d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{8}{3\,{a}^{4}d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-4\,{\frac{1}{{a}^{4}d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+8\,{\frac{1}{{a}^{4}d\sin \left ( dx+c \right ) }}+8\,{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{{a}^{4}d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^7/(a+a*sin(d*x+c))^4,x)

[Out]

-8*ln(1+sin(d*x+c))/a^4/d-1/6/d/a^4/sin(d*x+c)^6+4/5/d/a^4/sin(d*x+c)^5-7/4/d/a^4/sin(d*x+c)^4+8/3/d/a^4/sin(d
*x+c)^3-4/d/a^4/sin(d*x+c)^2+8/d/a^4/sin(d*x+c)+8*ln(sin(d*x+c))/a^4/d

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Maxima [A]  time = 2.68077, size = 128, normalized size = 0.95 \begin{align*} -\frac{\frac{480 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4}} - \frac{480 \, \log \left (\sin \left (d x + c\right )\right )}{a^{4}} - \frac{480 \, \sin \left (d x + c\right )^{5} - 240 \, \sin \left (d x + c\right )^{4} + 160 \, \sin \left (d x + c\right )^{3} - 105 \, \sin \left (d x + c\right )^{2} + 48 \, \sin \left (d x + c\right ) - 10}{a^{4} \sin \left (d x + c\right )^{6}}}{60 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^7/(a+a*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

-1/60*(480*log(sin(d*x + c) + 1)/a^4 - 480*log(sin(d*x + c))/a^4 - (480*sin(d*x + c)^5 - 240*sin(d*x + c)^4 +
160*sin(d*x + c)^3 - 105*sin(d*x + c)^2 + 48*sin(d*x + c) - 10)/(a^4*sin(d*x + c)^6))/d

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Fricas [A]  time = 1.59832, size = 502, normalized size = 3.72 \begin{align*} \frac{240 \, \cos \left (d x + c\right )^{4} - 585 \, \cos \left (d x + c\right )^{2} + 480 \,{\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) - 480 \,{\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 16 \,{\left (30 \, \cos \left (d x + c\right )^{4} - 70 \, \cos \left (d x + c\right )^{2} + 43\right )} \sin \left (d x + c\right ) + 355}{60 \,{\left (a^{4} d \cos \left (d x + c\right )^{6} - 3 \, a^{4} d \cos \left (d x + c\right )^{4} + 3 \, a^{4} d \cos \left (d x + c\right )^{2} - a^{4} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^7/(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

1/60*(240*cos(d*x + c)^4 - 585*cos(d*x + c)^2 + 480*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1)
*log(1/2*sin(d*x + c)) - 480*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1)*log(sin(d*x + c) + 1)
- 16*(30*cos(d*x + c)^4 - 70*cos(d*x + c)^2 + 43)*sin(d*x + c) + 355)/(a^4*d*cos(d*x + c)^6 - 3*a^4*d*cos(d*x
+ c)^4 + 3*a^4*d*cos(d*x + c)^2 - a^4*d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**7/(a+a*sin(d*x+c))**4,x)

[Out]

Timed out

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Giac [A]  time = 1.93045, size = 313, normalized size = 2.32 \begin{align*} -\frac{\frac{30720 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac{15360 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{4}} + \frac{37632 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 10080 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 2835 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 880 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 240 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 48 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 5}{a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6}} + \frac{5 \, a^{20} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 48 \, a^{20} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 240 \, a^{20} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 880 \, a^{20} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2835 \, a^{20} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 10080 \, a^{20} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{24}}}{1920 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^7/(a+a*sin(d*x+c))^4,x, algorithm="giac")

[Out]

-1/1920*(30720*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^4 - 15360*log(abs(tan(1/2*d*x + 1/2*c)))/a^4 + (37632*tan(
1/2*d*x + 1/2*c)^6 - 10080*tan(1/2*d*x + 1/2*c)^5 + 2835*tan(1/2*d*x + 1/2*c)^4 - 880*tan(1/2*d*x + 1/2*c)^3 +
240*tan(1/2*d*x + 1/2*c)^2 - 48*tan(1/2*d*x + 1/2*c) + 5)/(a^4*tan(1/2*d*x + 1/2*c)^6) + (5*a^20*tan(1/2*d*x
+ 1/2*c)^6 - 48*a^20*tan(1/2*d*x + 1/2*c)^5 + 240*a^20*tan(1/2*d*x + 1/2*c)^4 - 880*a^20*tan(1/2*d*x + 1/2*c)^
3 + 2835*a^20*tan(1/2*d*x + 1/2*c)^2 - 10080*a^20*tan(1/2*d*x + 1/2*c))/a^24)/d